This is an excellent historical and mathematical view by a renowned Italian geometer of the geometries that have risen from a rejection of Euclid's parallel postulate. Students, teachers and mathematicians will find here a ready reference source and guide to a field that has now become overwhelmingly important.
Non-Euclidean Geometry first examines the various attempts to prove Euclid's parallel postulate-by the Greeks, Arabs, and mathematicians of the Renaissance. Then, ranging through the 17th, 18th and 19th centuries, it considers the forerunners and founders of non-Euclidean geometry, such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, Schweikart, Taurinus, J. Bolyai and Lobachevski. In a discussion of later developments, the author treats the work of Riemann, Helmholtz and Lie; the impossibility of proving Euclid's postulate, and similar topics. The complete text of two of the founding monographs is appended to Bonola's study: "The Science of Absolute Space" by John Bolyai and "Geometrical Researches on the Theory of Parallels" by Nicholas Lobachevski. "Firmly recommended to any scientific reader with some mathematical inclination" — Journal of the Royal Naval Scientific Service. "Classic on the subject." — Scientific American.
Table of ContentsChapter I. The Attempts to prove Euclid's Parallel Postulate.
1-5. The Greek Geometers and the Parallel Postulate
6. The Arabs and the Parallel Postulate
7-10. The Parallel Postulate during the Renaissance and the 17th Century
Chapter II. The Forerunners on Non-Euclidean Geometry.
11-17. GEROLAMO SACCHERI (1667-1733)
18-22. JOHANN HEINRICH LAMBERT (1728-1777)
23-26. The French Geometers towards the End of the 18th Century
27-28. ADRIEN MARIE LEGENDRE (1752-1833)
29. WOLFGANG BOLYAI (1775-1856)
30. FRIEDRICH LUDWIG WACHTER (1792-1817)
30. (bis) BERNHARD FRIEDRICH THIBAUT (1776-1832)
Chapter III. The Founders of Non-Euclidean Geometry.
31-34. KARL FRIEDRICH GAUSS (1777-1855)
35. FERDINAND KARL SCHWEIKART (1780-1859)
36-38. FRANZ ADOLF TAURINUS (1794-1874)
Chapter IV. The Founders of Non-Euclidean Geometry (Cont.).
39-45. NICOLAI IVANOVITSCH LOBATSCHEWSKY (1793-1856)
46-55. JOHANN BOLYAI (1802-1860)
56-58. The Absolute Trigonometry
59. Hypotheses equivalent to Euclid's Postulate
60-65. The Spread of Non-Euclidean Geometry
Chapter V. The Later Development of Non-Euclidean Geometry.
Differential Geometry and Non-Euclidean Geometry
67-69. Geometry upon a Surface
70-76. Principles of Plane Geometry on the Ideas of RIEMANN
77. Principles of RIEMANN'S Solid Geometry
78. The Work of HELMHOLTZ and the Investigations of LIE
Projective Geometry and Non-Euclidean Geometry
79-83. Subordination of Metrical Geometry to Projective Geometry
84-91. Representation of the Geometry of LOBATSCHEWSKY-BOLYAI on the Euclidean Plane
92. Representation of RIEMANN'S Elliptic Geometry in Euclidean Space
93. Foundation of Geometry upon Descriptive Properties
94. The Impossibility of proving Euclid's Postulate
Appendix I. The Fundamental Principles of Statistics and Euclid's Postulate.
1-3. On the Principle of the Lever
4-8. On the Composition of Forces acting at a Point
9-10. Non-Euclidean Statics
11-12. Deduction of Plane Trigonometry from Statics
Appendix II. CLIFFORD'S Parallels and Surface. Sketch of CLIFFFORD-KLEIN'S Problems.
1-4. CLIFFORD'S Parallels
5-8. CLIFFORD'S Surface
9-11. Sketch of CLIFFORD-KLEIN'S Problem
Appendix III. The Non-Euclidean Parallel Construction and other Allied Constructions.
1-3. The Non-Euclidean Parallel Construction
4. Construction of the Common Perpendicular to two non-intersecting Straight Lines
5. Construction of the Common Parallel to the Straight Lines which bound an Angle
6. Construction of the Straight Line which is perpendicular to one of the lines bounding an acute Angle and Parallel to the other
7. The Absolute and the Parallel Construction
Appendix IV. The Independence of Projective Geometry from Euclid's Postulate.
1. Statement of the Problem
2. Improper Points and the Complete Projective Plane
3. The Complete Projective Line
4. Combination of Elements
5. Improper Lines
6. Complete Projective Space
7. Indirect Proof of the Independence of Projective Geometry from the Fifth Postulate
8. BELTRAMI'S Direct Proof of this Independence
Appendix V. The Impossibility of proving Euclid's Postulate. An Elementary Demonstration of this Impossibility founded upon the Properties of the System of Circles orthogonal to a Fixed Circle.
2-7. The System of Circles passing through a Fixed Point
8-12. The System of Circles orthogonal to a Fixed Circle
Index of Authors
The Science of Absolute Space and the Theory of Parallels___________________follow